How can I move a UV Island to simulate the Movement of an Eye
Ok... it seems it is needed to complete a previous answer (Creating an eye/eyelid rig for dragon without shape keys)...

You are more or less in this situation with a single UV map and a single texture. The eyes are here selected on the left.

On the right, you can see two vertex groups (named WarpLeftEye and WarpRightEye) : each of them is assigned the corresponding eye vertices.



Now, add two empties. Here I created a sphere and a cube one (in order to see who is who). Then setup the UV Warp modifier this way, using the two empties (or any objects or bones) and the vertex group for the left eye (do the same for the other eye) :



Here is the result :



Surely, you'll need to have enough space around the eyes in your texture. If not, add another one as suggested in the comments.
You can also use the UV map field of the modifier to use another UV map done for the eyes.

Note : scaling and rotating work too... more or less easy to do but simply scale and move and/or rotate the empty (scaling will shift too : so you'll need to scale a bit, realign, scale a bit, etc.)